Algebraically Compact Groups

In the preceding chapter we have encountered groups that were summands in every group containing them as pure subgroups: the pure-injective groups. In this chapter, we collect a large amount of additional information about these groups. Interestingly, these are precisely the summands of groups admitting a compact group topology, and the reduced ones are nothing else than the groups complete in the Z-adic topology. From Sect. 4 in Chapter 5 we know that every group can be embedded as a pure subgroup in a pure-injective (i.e., in an algebraically compact) group, and here we show that the significance of this embedding is enhanced by the fact that minimal embeddings exist and are unique up to isomorphism. Thus the theory of algebraically compact groups runs, in many respects, parallel to the theory of injective groups, a fact that was first pointed out by Maranda [1].The theory of algebraically compact groups is quite satisfactory: these groups admit complete characterization by cardinal invariants. We shall often meet algebraically compact groups in subsequent discussions.We close this chapter with the discussion of the exchange property. This is a remarkable, but rather rare phenomenon. Groups with this property show the best behavior as summands.